Cambridge Additional Mathematics

TANGENT

FUNCTION

DYNAMIC

TANGENT

FUNCTION

¼ ¼ ¼ ¼

¼

4

3

2 2

¼

4

• 3


• 
  

¼

4

¼

4

y

O x

y=tan_2x

-_-_¼¼-_-_E_f_E_fp_p -_-_wpp_w_ -_-_r_rp_p _p_rpr _pw_pw _EE_f_fpp_ ¼¼

y DEMO

3

-3

x

y=tanx

-_2-_2¼¼-_-_E_s_Es__pp -_-_¼¼ -_-_pwpw__ O p_pw_w ¼¼ E_sE_s_p_p 22 ¼¼ _TsT_s_p_p

Trigonometric functions (Chapter 9) 239

THE GRAPH OF y= tanx

Since tanx=

sinx

cosx

, tanxwill be undefined whenever cosx=0.

The zeros of the function y= cosx correspond to vertical asymptotes of the function y= tanx.

We observe that y= tanx has:

² period¼

² range y 2 R

² vertical asymptotes x=¼ 2 +k¼ for all k 2 Z.

Click on the icon to explore how the tangent function is produced from the unit circle.

THE GENERAL TANGENT FUNCTION

Thegeneral tangent functionis y=atanbx+c, a> 0 , b> 0.

² Theprincipal axisis y=c.

² Theperiodof this function is

¼

b

.

² Theamplitudeof this function is undefined.

Click on the icon to explore the properties of this function.

Example 4 Self Tutor

Without using technology, sketch the graph of y= tan 2x for ¡¼ 6 x 6 ¼.

Since b=2, the period is ¼ 2.

The vertical asymptotes are

x=§¼ 4 , x=§^34 ¼,

and thex-axis intercepts are at

0 ,§¼ 2 ,§¼.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_09\239CamAdd_09.cdr Monday, 6 January 2014 4:51:27 PM BRIAN